Dickens chose names for obvious and somewhat less obvious symbolism. In this way, a character’s name could characterize the character, and convey something important about the character.
Twist is a name given to Oliver when he is born, not his real name. Throughout the story, Oliver is constantly “twisted” around, as he is used by everyone he comes in contact with. Each of the other characters you mention use him.
Sowerberry of course makes one think of a “sour berry.” A berry is expected to be a good thing, and in fact Oliver is somewhat better off working as an undertaker’s assistant than in the workhouse. Soweberry is described with "features were not naturally intended to wear a smiling aspect, but he was in general rather given to professional jocosity" (enotes etext p. 16)
Indeed, he laughs at the idea of the poor dying and needing coffins. He is a bitter man. He notes that Oliver has a look of melancholy that would work for children’s funerals. Sowererry’s one good quality is that he is not terribly abusive.
To do him justice, he was, as far as his power went--it was not very extensive--kindly disposed towards the boy; perhaps, because it was his interest to be so; perhaps, because his wife disliked him. (p. 33)
But he beats Oliver anyway, because his wife and Bumble would think ill of him if did not.
Mrs. Mann is the caretaker of the house where Oliver stays. She is cruel and abusive, using the children to make her fortune. She starves and beats them, and takes money for them. She makes a pretense of washing them when Bumble comes, but he is complicit anyway. When Bumble describes her, the lines drip with satire: he says she is a "humane woman" and "like a mother" (p. 6).
So Mrs. Mann’s name represents her NOT acting motherly. She acts with the more “manly” qualities of greed and tyranny.
Monks’s name is also ironic. A monk is someone you would expect to be kind, reserved, and religious. Monks feels victimized by Oliver. He is none of these things. He seeks only to destroy the boy who is his half-brother, and asks why Fagin has not '"made a sneaking, snivelling pickpocket of him" (p. 130)
He wants Fagin to destroy his brother by turning him into a criminal. This way, he can keep his father’s fortune. He is a criminal, and seems to have no affection for the boy.
The Artful Dodger is another nickname given to Jack Dawkins. He’s clever and sneaky and very good at his job of pickpocket. His name is a nickname, but it also goes right to the issue of his personality. He can “dodge” his way out of any situation. Dodger does use Oliver, but seems to show some affection for him to. When Charley Bates calls Oliver “green”, “The Dodger said nothing, but he smoothed Oliver's hair over his eyes, and said he'd know better, by and by” (p. 42).
Dodger also seems adept at handling Fagin. When he returns without Oliver, he knows Fagin is going to be angry. Fagin grabs him and shakes him, coat and all.
Why, the traps have got him, and that's all about it,' said the Dodger, sullenly. 'Come, let go o' me, will you!' And, swinging himself, at one jerk, clean out of the big coat, which he left in the Jew's hands, the Dodger snatched up the toasting fork, and made a pass at the merry old gentleman's waistcoat. (p. 58)
At that point Sikes enters and the conversation shifts, but it shows that Dodger lives a hard life and has learned to think on his feet.